Here are two possibilities to get the precision you want:
1. Use exact input:
In: N[(5/2)^125, 30]
Out: 5.52714787526044456024726519219*10^49
2. Use a more precise input:
In: 2.500000000000000000000000000000^125]
Out: 5.527147875260444560247265192*10^49
There's probably a more elegant way to do this, but being pretty new at Mathematica myself, those were the only things I could think of.